When the polynomial matrix \(P(\lambda )\)  has real coefficients \(P_i\) , these conditions become conditions on the transpose of each \(P_i\) , and the polynomial matrices are said to be symmetric, skew-symmetric, para-symmetric and para-skew-symmetric. We emphasize that the nomenclature above is the one commonly used for structured rational matrices in the linear system and control theory literature (see, for instance, [1]}, [2]}, [3]} and the references therein). However, in the literature focused on polynomial matrices, para-Hermitian and para-symmetric polynomial matrices are called \(*\) -even and T-even, respectively, while para-skew-Hermitian and para-skew-symmetric polynomial matrices are called \(*\) -odd and T-odd, respectively [4]}, and all of them are called generically alternating polynomial matrices [5]}.
